Schmidt Decomposable Products of Projections

Schmidt Decomposable Products of Projections
Andruchow, Esteban; Corach, Gustavo
2017-10-11 00:00:00
We characterize operators
$$T=PQ$$
T
=
P
Q
(P, Q orthogonal projections in a Hilbert space
$${\mathcal {H}}$$
H
) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases
$$\{\psi _n\}$$
{
ψ
n
}
of R(P) and
$$\{\xi _n\}$$
{
ξ
n
}
of R(Q) such that
$$\langle \xi _n,\psi _m\rangle =0$$
⟨
ξ
n
,
ψ
m
⟩
=
0
if
$$n\ne m$$
n
≠
m
. Also it is shown that this is equivalent to
$$A=P-Q$$
A
=
P
-
Q
being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener–Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if
$$T=PQ$$
T
=
P
Q
has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngIntegral Equations and Operator TheorySpringer Journalshttp://www.deepdyve.com/lp/springer-journals/schmidt-decomposable-products-of-projections-04JMR7wHXy

Abstract

We characterize operators
$$T=PQ$$
T
=
P
Q
(P, Q orthogonal projections in a Hilbert space
$${\mathcal {H}}$$
H
) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases
$$\{\psi _n\}$$
{
ψ
n
}
of R(P) and
$$\{\xi _n\}$$
{
ξ
n
}
of R(Q) such that
$$\langle \xi _n,\psi _m\rangle =0$$
⟨
ξ
n
,
ψ
m
⟩
=
0
if
$$n\ne m$$
n
≠
m
. Also it is shown that this is equivalent to
$$A=P-Q$$
A
=
P
-
Q
being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener–Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if
$$T=PQ$$
T
=
P
Q
has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent.

Journal

Integral Equations and Operator Theory
– Springer Journals

Published: Oct 11, 2017

Recommended Articles

Loading...

References

Classes of idempotents in Hilbert space

Andruchow, E

Operators which are the difference of two projections

Andruchow, E

Hilbert space idempotents and involutions

Buckholtz, D

Polar decomposition of oblique projections

Corach, G; Maestripieri, A

Products of orthogonal projections and polar decompositions

Corach, G; Maestripieri, A

The geometry of spaces of projections algebras

Corach, G; Porta, H; Recht, L

Separation of two linear subspaces

Davis, C

Uncertainty principles and signal recovery

Donoho, DL; Stark, PB

The uncertainty principle: a mathematical survey

Folland, GB; Sitaram, A

Solutions of the matrix equation
$$XAX=X$$
X
A
X
=
X
, and relations between oblique and orthogonal projectors